Theorem preq1b

Description: Biconditional equality lemma for unordered pairs, deduction form. Two unordered pairs have the same second element iff the first elements are equal. (Contributed by AV, 18-Dec-2020)

	Ref	Expression
Hypotheses	preq1b.a	|- ( ph -> A e. V )
	preq1b.b	|- ( ph -> B e. W )
Assertion	preq1b	|- ( ph -> ( { A , C } = { B , C } <-> A = B ) )

Proof

Step	Hyp	Ref	Expression
1		preq1b.a	|- ( ph -> A e. V )
2		preq1b.b	|- ( ph -> B e. W )
3		prid1g	|- ( A e. V -> A e. { A , C } )
4	1 3	syl	|- ( ph -> A e. { A , C } )
5		eleq2	|- ( { A , C } = { B , C } -> ( A e. { A , C } <-> A e. { B , C } ) )
6	4 5	syl5ibcom	|- ( ph -> ( { A , C } = { B , C } -> A e. { B , C } ) )
7		elprg	|- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
8	1 7	syl	|- ( ph -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
9	6 8	sylibd	|- ( ph -> ( { A , C } = { B , C } -> ( A = B \/ A = C ) ) )
10	9	imp	|- ( ( ph /\ { A , C } = { B , C } ) -> ( A = B \/ A = C ) )
11		prid1g	|- ( B e. W -> B e. { B , C } )
12	2 11	syl	|- ( ph -> B e. { B , C } )
13		eleq2	|- ( { A , C } = { B , C } -> ( B e. { A , C } <-> B e. { B , C } ) )
14	12 13	syl5ibrcom	|- ( ph -> ( { A , C } = { B , C } -> B e. { A , C } ) )
15		elprg	|- ( B e. W -> ( B e. { A , C } <-> ( B = A \/ B = C ) ) )
16	2 15	syl	|- ( ph -> ( B e. { A , C } <-> ( B = A \/ B = C ) ) )
17	14 16	sylibd	|- ( ph -> ( { A , C } = { B , C } -> ( B = A \/ B = C ) ) )
18	17	imp	|- ( ( ph /\ { A , C } = { B , C } ) -> ( B = A \/ B = C ) )
19		eqcom	|- ( A = B <-> B = A )
20		eqeq2	|- ( A = C -> ( B = A <-> B = C ) )
21	10 18 19 20	oplem1	|- ( ( ph /\ { A , C } = { B , C } ) -> A = B )
22	21	ex	|- ( ph -> ( { A , C } = { B , C } -> A = B ) )
23		preq1	|- ( A = B -> { A , C } = { B , C } )
24	22 23	impbid1	|- ( ph -> ( { A , C } = { B , C } <-> A = B ) )